5c. Solve the differential equations using Laplace transforms
y"+y' - 2y = e-³x, y(0) = 0 and y'(0) = 0
Answers
Answer:
[tex]y(x)=\frac{1}{2}e^{-3x}-\frac{2}{3}e^{-2x}+\frac{1}{6}e^{x}[/tex]
Step-by-step explanation:
[tex]y''+y'-2y=e^{-3x},\: y(0)=0,\: y'(0)=0\\\\\mathcal{L}\{y''\}+\mathcal{L}\{y'\}-2\mathcal{L}\{y\}=\mathcal{L}\{e^{-3x}\}\\\\s^2Y(s)-sy(0)-y'(0)+sY(s)-y(0)-2Y(s)=\frac{1}{s+3} \\ \\s^2Y(s)+sY(s)-2Y(s)=\frac{1}{s+3}\\ \\(s^2+s-2)Y(s)=\frac{1}{s+3}\\ \\Y(s)=\frac{2}{(s+3)(s^2+s-2)}\\ \\Y(s)=\frac{2}{(s+3)(s+2)(s-1)}[/tex]
Perform the partial fraction decomposition
[tex]\frac{2}{(s+3)(s+2)(s-1)}=\frac{A}{s+3}+\frac{B}{s+2}+\frac{C}{s-1}\\ \\2=(s+2)(s-1)A+(s+3)(s-1)B+(s+2)(s+3)C[/tex]
Solve for each constant
[tex]2=((-2)+2)((-2)-1)A+((-2)+3)((-2)-1)B+((-2)+2)((-2)+3)C\\\\2=-3B\\\\-\frac{2}{3}=B[/tex]
[tex]2=(1+2)(1-1)A+(1+3)(1-1)B+(1+2)(1+3)C\\ \\2=12C\\\\\frac{2}{12}=C\\ \\\frac{1}{6}=C[/tex]
[tex]2=((-3)+2)((-3)-1)A+((-3)+3)((-3)-1)B+((-3)+2)((-3)+3)C\\ \\2=4A\\\\\frac{2}{4}=A\\ \\\frac{1}{2}=A[/tex]
Take the inverse transform and solve for the IVP
[tex]Y(s)=\frac{\frac{1}{2}}{s+3}+\frac{-\frac{2}{3}}{s+2}+\frac{\frac{1}{6}}{s-1}\\\\y(x)=\frac{1}{2}e^{-3x}-\frac{2}{3}e^{-2x}+\frac{1}{6}e^{x}[/tex]
Related Questions
find the x-intercept y=(x-2)/(3x-1)
Answers
Answer:
y=-2
x=-6
Step-by-step explanation:
What is the product of 22 and 49
Answers
Answer:
1078
Step-by-step explanation:
Hope this helps
Use the graph of the function to estimate the intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
Answers
The intervals on which the graph is increasing: ]-∞,-3[ U ]0.5,-∞[ . On the other hand, the graph is decreasing: ]-3,0.5[
FunctionA function can be classified as increasing or decreasing. Thus, a function is increasing when the y-values increase, on the other hand, a function is decreasing when the y-values decrease.
From the image, you can see that the y-values increase in the following x-intervals: from -∞ to -3 and from 0.5 to ∞. Using interval notation, you can write that the function is increasing in:
]-∞,-3[ U ]0.5,-∞[
From the graph, you can see that the y-values decrease in the following x-intervals: from -3 to 0.5. Using interval notation, you can write that the function is decreasing in:
]-3,0.5[
Learn more about function here:
https://brainly.com/question/2649645
Twice the difference of a number and 2 equals 5
Answers
When something says twice a number it’s always going to be multiplied by 2 and “a number will always be a variable
9514 1404 393
Answer:
2(x -2) = 5
Step-by-step explanation:
If x represents the number, then the difference of the number and 2 is ...
x -2
Twice the difference is ...
2(x -2)
"equals 5" means that product is equal to 5. The appropriate equation is ...
2(x -2) = 5
_____
In case you want a solution ...
Dividing by 2, we get ...
x -2 = 5/2
Adding 2 gives the value of the number:
x = 2 +5/2 = 9/2
x = 4 1/2
The number is 4 1/2.
FIRST TO ANSWER 50 POINTS !!
Randy and Carla like to walk the path around their town park. The path is 2 miles long. Last month Randy walked the path 13 times, and Carla walked it 22 times. How many more miles did Carla walk than Randy last month?
Answers
Answer: 18 more miles
Step-by-step explanation:
5. Write the equation in slope-intercept form that
is parallel to y = 4x - 2 and goes through the
point (-2, 3).
Answers
Answer:
y=4x+11
Step-by-step explanation:
Same slope because the lines are parallel
A bag contains five Snickers bars, nine Milky Way bars, two Crunch bars,
and twelve Twix bars. If Jack randomly selects a candy bar, eats it, then
randomly selects another, find the probability that both candy bars were
Milky Way.
Answers
Answer:
about a 9.5 percent chance of both being a milky way
Step-by-step explanation:
Probability that both candy bars were milky way is [tex]\frac{2}{21}[/tex]
What is Probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur
Given,
Number of snickers bars = 5
Number of Milky way bars =9
Number of crunch bars = 2
Number of Twix bars = 12
Total number of Candy bars = 5+9+2+12 =28
If Jack randomly selects a candy bar, eats it, then randomly selects another
Probability = Number of favorable outcome / Number of total outcome
Probability of taking Milky way bar in first pick = [tex]\frac{9}{28}[/tex]
So Total number of candy bars and number of Milky way bar are decreased by 1.
Probability of taking Milky way bar in second pick = [tex]\frac{8}{27}[/tex]
Probability that both candy bars were milky way = [tex](\frac{9}{28}).(\frac{8}{27})=\frac{2}{21}[/tex]
Hence, Probability that both candy bars were milky way is [tex]\frac{2}{21}[/tex]
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What is the slope of the line passing through the points (4, -3) and (1, -3)? *
Answers
Answer:
[tex]\huge\boxed{\sf Slope = 0}[/tex]
Step-by-step explanation:
[tex]\sf Slope = \frac{rise}{run} \\\\Slope = \frac{y2-y1}{x2-x1} \\\\Given\ the\ coordinates\ (4,-3) \ and \ (1,-3)\\\\ x1 = 4 ,\ y1 = -3 ,\ x2 = 1 ,\ y 2 = -3\\\\So,\\\\Slope = \frac{-3-(-3)}{1-4} \\\\Slope = \frac{0}{-3} \\\\Slope = 0\\\\\rule[225]{225}{2}[/tex]
Hope this helped!
~AnonymousHelper1807Which statement is false?
0.8 = 3/5
2/3 > 0.34
2.375 > 2 1/4
3 8/11 < 3 12/13
Answers
Answer:
Step-by-step explanation:
0.8 = 3/5
3/5 = 6/10 = .6
FALSE
2/3 > 0.34
2/3 = .333333...
FALSE
2.375 > 2 1/4
2 1/4 = 2.25
TRUE
3 8/11 < 3 12/13
3 8/11 = 3.727272
3 12/13 = 3.9231
TRUE
Writing Linear Equations
Answers
The answer to A is Gabe, the answer to B is Mila and I don't have an answer to C, try inputting it into a calculator. Sorry.
A)
y = ax + b
b = 40 as the line starts at (0 ; 40)
then the line goes down => so a < 0
between the 1st point and the second, we go 2 units on the right and we go 1 unit down
so a = - 1/2 = - 0,5
finally y = - 0,5x + 40
B)
y = ax + b
with
a = (-6 ; 0) / (0 ; -30) = 6/30 = 1/5
with the point ( 0 ; -6) we know that b = -6
finally y = 1/5x - 6
C)
we use the point (0 ; - 10)
so Amir is right as b will be - 10
PLS HELP ME !
Draw out a two column proof for each problem below.
Make sure to start with your given statement and write out each statement and reason.
Answers
Answer and Explanation:
See attached image.
The distance, in feet, two boys travel per second on a treadmill is shown to the left. Which comparison is accurate?
Xavier is traveling at 1.5 feet per second.
Moises is going faster than Xavier.
The difference in their rates of change is 1 foot per second.
If both boys remain on the treadmill for 10 minutes, Xavier will have traveled a greater distance. (a straight up answer)
Answers
Answer:
Step-by-step explanation:
D is correct
Answer: D
Step-by-step explanation:
I need help please
Answers
Answer:
C
Step-by-step explanation:
A. 3:7 is not equal to 5:14 because both of these options are in their lowest value and are not equal.
B. 8:6 is not equal to 6:8 because 8 is not equal to 6 and 6 is not equal to 8.
C. 1:4 is equal to 5:20 because 1:4 can be multiplied by 5 to be 5:20 and 5:20 is equal to 5:20.
D. 5:3 is not equal to 12:5 because 5 is not equal to 12 and 3 is not equal to 15.
Hope this Helps! :)
Have any questions? Ask below in the comments and I will try my best to answer.
-SGO
What is 70 percent of 20?
Answers
Answer:
14
Step-by-step explanation:
All you need to do is turn 70% into a decimal which is .70
Then multiply .70 and 20
.70*20
which gives you 14
The number is 14, which represents 70 percent of 20 is 14.
As we know that a percentage is normally calculated by dividing the component by the total and then multiplying by 100.
To find 70 percent of a number, we multiply the number by 70/100 or 0.7.
As per the question, we want to find 70 percent of 20.
So, we have:
70% × 20 = (70/100) × 20
To simplify the calculation, we can convert the percentage to a decimal by dividing 70 by 100, which gives us 0.7:
(70/100) × 20 = 0.7 × 20
Multiplying 0.7 by 20, we get:
0.7 × 20 = 14
Therefore, 70 percent of 20 is 14.
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Enter any value of b that will make the following expression factorable
x2 + bx + 13
Answers
Answer:
14
Step-by-step explanation:
?*?=13 and ?+?=b
since 13*1=13, you could replace b with 14, since 13+1=14.
this will factor out into (x+1)(x+13)
What is the median of the data displayed in this box-and-whisker plot?
84.0
99.5
113.0
143.0
Answers
Answer: 113
Step-by-step explanation:
The median is the dot with the line through it in the middle of the box: so it is 113
You move 4 units forward, then 9 units backwards on a board game. Which of the following expressions can be used to find the sum of your moves?
4 + 9
4 + (-9)
-4 + 9
-4 + (-9)
Answers
Answer:
I believe the correct answer is option 2
4 + (-9)
This expresses your answer as a sum (addition) while accurately representing your moves (4 forward, being positive, and 9 backwards, being negative)
I have a question?.....Ms. Wallace makes soap to sell at craft fairs. The first bar of soap sells for $2, and each additional bar costs $1.25. The cost of making the soap is $0.95 per bar. Ms. Wallace pays $50 to rent a booth at the fair. How many bars of soap, b, will Ms. Wallace need to sell to at least break even?
Answers
Answer:
Has your teacher given you notes to practice?
Step-by-step explanation:
If no, then break the question down into pieces and write down what you know.
Write the slope-intercept form of an equation for a line with y-intercept -5 and slope 2.\
Answers
Answer:
y = -5x + 2
Step-by-step explanation:
CAN SOMEONE PLEASEEE HELP ME!!!!???
Answers
Answer:32%
Step-by-step explanation:
Problem 8-21 (Algorithmic)
Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:
Rental Class
Room Super Saver Deluxe Business
Type I $38 $43 —
Type II $16 $28 $45
Type I rooms do not have wireless Internet access and are not available for the Business rental class.
Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 140 rentals in the Super Saver class, 50 rentals in the Deluxe class, and 40 rentals in the Business class. Round Tree has 110 Type I rooms and 110 Type II rooms.
Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types.
Answers
Answer:
Step-by-step explanation:
Problem 8-21 (Algorithmic)
Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:
Rental Class
bella desire and javon collect stamps. desire has 5 more stamps than bella, and javon has 3 times as many stamps as desire altgether they have 200 stamps altogether
Answers
Stamps with desire =5+x
Stamps with Javon=3(5+x)
=15+3x
Total no of stamps=200
So;
200=x+5+x+15+3x
200=5x+20
200-20=5x
180/5=x
x=36
Therefore;
The no of stamps with Bella =36
No of stamps with desire=5+36
=41
No of stamps with javon=15+3(36)
=123
Verification;
36+41+123=200
Answer:
I answered the stupid question and you deleted it
Step-by-step explanation:
What is the y-intercept of the function f(x) = -2/9x + 1/3
Answers
Answer:
1/3
Step-by-step explanation:
It’s in the form y=mx+b
b is the y intercept
Quadrilaterals ABCDABCDA, B, C, D and KLMNKLMNK, L, M, N are congruent. Which of the following sequences of transformations maps KLMNKLMNK, L, M, N onto ABCDABCDA, B, C, D? Sequence A Sequence B A translation along the directed line segment \overline{MA} MA start overline, M, A, end overline, then a reflection over the horizontal line through point AAA. A reflection over the vertical line through point NNN, then a translation along the directed line segment \overline{NB} NB start overline, N, B, end overline. Choose 1 answer: Choose 1 answer:
Answers
Answer:
D
Step-by-step explanation:
I did it and got it right on khan
Answer:
D, Neither
Step-by-step explanation:
Which expression represents the product of
64 and x?
A. 64-X
B. 64 + x
C. 64x
D. 64/x
Answers
Product = Multiplication
4. Given a = 10.2, A = 122 degrees, and b = 6.8, use the Law of Sines to solve for B. Round your answer to the nearest degree.
5. Given a = 15, A = 36 degrees, and B = 70. use the Law of Sines to solve for b. Round your answer to the nearest whole number.
6. Given a = 9, b = 4, and c = 7, use the Law of Cosines to solve for A. Round your answer to the nearest degree.
Answers
Answer:
4) 34° (nearest degree)
5) 24 (nearest whole number)
6) 107° (nearest degree)
Step-by-step explanation:
Please see the attached picture for the full solution.
Q6: 107° is the angle in the 2nd quadrant.
(180° -73.398°≈ 107°)
Picture 2: Law of Sines
A scoop of ice cream in the shape of a whole sphere sits in a right cone. The radius of the ice cream scoop is 3.0 cm and the radius of the cone is 3.0 cm. a. What is the volume of the scoop of ice cream? Show all your work. b. How tall must the height of the cone be to fit all the ice cream without spilling if it melts? Show all your work.
Please help me
Answers
Answer:
a. 462
b. 49
Step-by-step explanation:
take pie as 22/7
a. volume= π r² h/3
22/7 × 3² × h/3
22/7 × 3 × h
22/7×3
cross multiply
462
b. h= 3 × v/π r²
3 × 462 over 22/7 × 3²
multiply each term by 7
7×3×7×462over22/7×3²
21×462over22×3²
21/1×462/198
= 9702/198
= 49
According to a Los Angeles Times study of more than 1 million medical dispatches from to , the response time for medical aid varies dramatically across Los Angeles (LA Times website, November 2012). Under national standards adopted by the Los Angeles Fire Department, rescuers are supposed to arrive within six minutes to almost all medical emergencies. But the Times analysis found that in affluent hillside communities stretching from Griffith Park to Pacific Palisades, firefighters failed to hit that mark nearly of the time. The following data show the response times, in minutes, for emergency calls in the Griffith Park neighborhood.
11.8 10.3 10.7 10.6 11.5 8.3 10.5 10.9 10.7 11.2
Based on this sample of ten response times, compute the following descriptive statistics:
a. Mean, median, and mode
b. Range and standard deviation
c. Should the response time of 8.3 minutes be considered an outlier in comparison to the other response times?
Answers
Answer:
A)Mean :10.65
Median =10.7
Mode : 10.7
B)Range = 3.5
Standard deviation :0.89916
C)The response time of 8.3 minutes should be considered an outlier in comparison to the other response times
Step-by-step explanation:
A)
Data : 11.8 ,10.3, 10.7, 10.6, 11.5, 8.3, 10.5, 10.9, 10.7, 11.2
[tex]Mean = \frac{\text{Sum of all observations}}{\text{No. of observations}}\\Mean = \frac{11.8 +10.3+ 10.7+ 10.6+ 11.5+ 8.3+ 10.5+ 10.9+ 10.7+ 11.2}{10}[/tex]
Mean = 10.65
Median: The mid value of the data
Data in ascending order
8.3
10.3
10.5
10.6
10.7
10.7
10.9
11.2
11.5
11.8
n=10(even)
[tex]Median = \frac{(\frac{n}{2})\text{th term}+(\frac{n}{2}+1)\text{th term}}{2}\\Median = \frac{(\frac{10}{2})\text{th term}+(\frac{10}{2}+1)\text{th term}}{2}\\Median = \frac{10.7+10.7}{2}\\Median = 10.7[/tex]
Mode : the most occurring frequency
10.7 is occurring twice while others are occurring once
So, Mode is 10.7
B) Range = Maximum - Minimum=11.8-8.3=3.5
Standard deviation : [tex]\sqrt{\frac{\sum(x-\bar{x})^2}{n}}[/tex]
Standard deviation :[tex]\sqrt{\frac{(11.8-10.65)^2+(10.3-10.65)^2+......+(10.9-10.65)^2+(10.7-10.65)^2+(11.2-10.65)^2}{10}}[/tex]
Standard deviation :0.89916
C)
8.3
,10.3
,10.5
,10.6
,10.7
,10.7
,10.9
,11.2
,11.5
,11.8
For Q1 ( Median of lower quartile )
8.3
,10.3
,10.5
,10.6
,10.7
[tex]Median = \frac{n+1}{2}\text{th term} =\frac{5+1}{2}=3 \text{rd term}=10.5[/tex]
For Q3( Median of Upper quartile )
10.7
,10.9
,11.2
,11.5
,11.8
[tex]Median = \frac{n+1}{2}\text{th term} =\frac{5+1}{2}=3 \text{rd term}=11.2[/tex]
IQR = Q3-Q1=11.2-10.5=0.7
Range :(Q1-1.5IQR, Q3-1.5IQR)
Range :[tex](10.5-1.5 \times 0.7, 11.2-1.5 \times 0.7)[/tex]
Range :(9.45, 10.15)
8.3 does not lie in this interval
So, the response time of 8.3 minutes should be considered an outlier in comparison to the other response times
What Is -5 1/2 x -1 1/8
Answers
✨Answer:✨
-6 3/16
✨Step-by-step explanation:✨
-5 1/2 x -1 1/8 = ?
Convert mixed numbers into improper fractions
-11/2 x -9/8 = -99/16 = -6 3/16
ANYONE I NEED HELP ASAP!!!!!